I am trying to model a well-known delay measurement algorithm in LTE Prach. Let me put it in simpler terms. The mobile user (UE) generates a signal of length 864 samples (ZC preamble). It maps them to 864 adjacent bins of 24K ifft. The resulting time domain signal is then sampled out at LTE sampling rate (let us call the 20MHz lte sample period as Ts).
The receiver (eNB) then applies the reverse i.e. 24k fft and extracts the 864 bins discarding all other bins. The extracted preamble is then correlated with a reference zero delay preamble same as that which was used by UE.
Accordingly the physical delay is in time domain in Ts units but correlation of preamble is in frequency domain in some ? units.
Based on my modelling the resolution of delay is about 28 Ts as I get one position change of correlation for every 28 Ts. My conclusion is just that; Prach algorithm resolution is 28 Ts .
I can't see any direct reference to delay resolution except that some saying the TA resolution is 16 Ts where TA is meant to be Timing Advance figure sent to that UE. Hence I view it as negative of delay resolution but this conflicts with my result of 28 Ts.
My immediate conclusion is that 28 comes from 24k/864 = ~ 28 (for 20MHz LTE case)
My view is that if original signal is 864 frequency bins, spread thin to 24k time domain samples and here is when delay occurs, then the 864 bins are extracted back, then how come the ratio can be better than 28. How can delay information in Ts units pass down to the 864 bins.
Though I notice in my model that delays from 16 Ts up to 28 Ts may be causing fractional delay effect at correlation but hard to quantify.
Any thoughts please?
I think you may have your IFFT length wrong.
1 Ts is 1/30.72e6, and the 20 MHz band is assumed to have a symbol length of 2048 samples, and so is the length of the FFT.
Shows that for the 20 MHz bandwidth, 1200 sub-carriers of the available 2048 are used.
I have not looked at the details, but it seems completely reasonable, that 864 of the 1200 are used for PRACH.
Maybe this will solve your problem.
Also, it is quite normal for the TA to be quantized to multiples of 16 Ts.
This is accurate enough to guarantee that the multi-path spread falls inside the cyclic prefix time of the symbols.
This is prach channel, not 2k fft lte data (pusch) case. it has spacing of 1.25KHz instead of 15KHz.
Prach is generated as I explained and then added to LTE having its own bins.
OK. I stand corrected. Thanks.
The Prach will be very long in time domain. In practice the 24K fft/ifft is managed in a different way (up/dn sampling) but the idea is simpler to view it as full fft size.
Discussions do help. It seems that we can target better resolution by upsampling before correlation.
If we ignore specifics of LTE prach and see it through principles of dsp what we are doing is this: have a signal at low sampling rate, upsample by 24, pass into channel (delay occurs), downsample by 24 and correlate to measure delay.
If we have to be fair with dsp nature we need to upsample back by anything up to 24 before we correlate.